Page 72 - Algorithm Collections for Digital Signal Processing Applications using MATLAB

60 Chapter 2

In other words, If the Columns of the matrix [B] are orthonormal to each
T
T
-1
other (i.e) [B] = [B] , then E[XX ]=[I] (i.e) Covariance Matrix of the
independent signals is the Identity Matrix.(Required).
-1
[A][Z]={[B][U][B]} [X]
When [A] is estimated such that covariance matrix of the LHS and RHS are
-1
Identity Matrix, [A][Z]=[X] (ie) {[B][U][B]} becomes the Identity matrix.

Thus ICA Problem is the optimization problem as mentioned below.

Given the matrix [Y], estimating the best values for the matrix [B] such

that kurtosis measured for the row vectors of the matrix [X] =
T
[A][Z]=[B ][Z] is maximum.
T
Subject to the constraint B B=[I] (i.e) column vectors of the matrix B is
orthonormal to each other.

Let [B] = b 11 b 12
b 21 b 22

Z = z 11 z 12 z 13 … z 1n

z 21 z 22 z 23 … z 2n

T
[B] [Z] = b 11 b 21 z 11 z 12 z 13 … z 1n
b 12 b 22 z 21 z 22 z 23 … z 2n

= b 11z 11+b 21z 21 b 11z 12+b 21z 22 … b 11z 1n+b 21z 2n

b 12z 11+b 22z 21 b 12z 12+b 22z 22 … b 12z 1n+b 22z 2n

2
2
4
Kurtosis of the first row = E[(b 11z 1i+b 21z 2i ) ] – 3 {E[(b 11z 1i+b 21z 2i ) ]}
2
2
4
Kurtosis of the second row= E[(b 12z 1i+b 22z 2i ) ] – 3 {E[(b 12z 1i+b 22z 2i ) ]}